Create Equations

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Create Equations · Grade 9 · Worksheet 1

1. Ava is designing a triangular garden where the area is 84 square meters. The base is 7 meters more than the height. Write an equation relating the area to the base and height. Answer: A = (1/2)bh where b = h + 7 Solution: The area of a triangle is calculated as one-half times base times height. When one dimension is expressed in terms of the other, we can create an equation with two variables that represents their relationship.
2. A scientist is studying bacterial growth in a lab culture. The population P(t) after t hours is modeled by the exponential function P(t) = 500 × 2^(t/3). If the scientist needs the population to reach exactly 4,000 bacteria to begin an experiment, how many hours will it take for the culture to reach this population? Answer: 9 Solution: Step 1: Set up the equation using the given function: 500 × 2^(t/3) = 4000 Step 2: Divide both sides by 500: 2^(t/3) = 8 Step 3: Recognize that 8 is a power of 2: 2^(t/3) = 2^3 Step 4: Since the bases are equal, set the exponents equal: t/3 = 3 Step 5: Multiply both sides by 3: t = 9 Step 6:…
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 500 × 2^(t/3) = 4000 Step 2: Divide both sides by 500: 2^(t/3) = 8 Step 3: Recognize that 8 is a power of 2: 2^(t/3) = 2^3 Step 4: Since the bases are equal, set the exponents equal: t/3 = 3 Step 5: Multiply both sides by 3: t = 9 Step 6: Verify by substituting back: P(9) = 500 × 2^(9/3) = 500 × 2^3 = 500 × 8 = 4000 The answer is 9 hours.

3. Matiu is planning a rectangular vegetable garden. The length of the garden is 6 meters more than twice its width. The total area of the garden is 216 square meters. Write an equation in two variables that represents this situation, then determine the width and length of the garden. Answer: Width = 9 meters, Length = 24 meters Solution: Let w represent the width in meters and l represent the length in meters. The length is 6 meters more than twice the width: l = 2w + 6. The area of a rectangle is A = l * w.
   Full step-by-step solution

   Step 1: Let w represent the width in meters and l represent the length in meters. Step 2: The length is 6 meters more than twice the width: l = 2w + 6. Step 3: The area of a rectangle is A = l * w. Substitute the given area and the expression for l: 216 = (2w + 6) * w. Step 4: Expand: 216 = 2w^2 + 6w. Step 5: Rearrange to standard quadratic form: 2w^2 + 6w - 216 = 0. Step 6: Divide the entire equation by 2 to simplify: w^2 + 3w - 108 = 0. Step 7: Factor the quadratic: (w + 12)(w - 9) = 0. Step 8: Set each factor to zero: w + 12 = 0 gives w = -12 (not possible for width), w - 9 = 0 gives w = 9. Step 9: So the width is 9 meters. Then length = 2(9) + 6 = 18 + 6 = 24 meters. Step 10: Verify area: 9 * 24 = 216 square meters, which matches. The width is 9 meters and the length is 24 meters.

4. Ava is designing a rectangular banner for a school event. The area of the banner is 81 square feet, and the length is 6 feet more than the width. Write an equation in terms of width w and length l that represents this situation. Answer: l = w + 6, l × w = 81 Solution: When creating equations from geometric contexts, identify the given relationships between variables and the formulas that connect them.
   Full step-by-step solution

   When creating equations from geometric contexts, identify the given relationships between variables and the formulas that connect them. For rectangles, area equals length times width, and additional constraints can create systems of equations.

5. Mere is planning a rectangular vegetable garden. The length of the garden is 7 meters more than three times its width. The perimeter of the garden is 94 meters. Write a system of equations in two variables to represent this situation, and then determine the length and width of the garden. Answer: width = 10 meters, length = 37 meters Solution: Let w represent the width (in meters) and l represent the length (in meters). Step 2: From the problem, the length is 7 meters more than three times the width, so l = 3w + 7.
   Full step-by-step solution

   Step 1: Let w represent the width (in meters) and l represent the length (in meters). Step 2: From the problem, the length is 7 meters more than three times the width, so l = 3w + 7. Step 3: The perimeter of a rectangle is given by P = 2l + 2w, and we know P = 94, so 2l + 2w = 94. Step 4: Substitute the expression for l from Step 2 into the perimeter equation: 2(3w + 7) + 2w = 94. Step 5: Simplify: 6w + 14 + 2w = 94, which gives 8w + 14 = 94. Step 6: Subtract 14 from both sides: 8w = 80. Step 7: Divide by 8: w = 10. Step 8: Substitute w = 10 back into l = 3w + 7: l = 3(10) + 7 = 30 + 7 = 37. Step 9: The width is 10 meters and the length is 37 meters. Check: Perimeter = 2(37) + 2(10) = 74 + 20 = 94 meters, which matches. The answer is width = 10 meters, length = 37 meters.

6. Kaia is organizing a fundraising concert. She rents a venue that costs a fixed fee of $450. Additionally, she needs to pay for security guards, where each guard costs $85, and for sound equipment, which costs a flat rate of $220 plus $15 per hour of use. Write an equation in two or more variables that represents the total cost C of the event in terms of the number of security guards g and the number of hours h the sound equipment is used. Answer: C = 85g + 15h + 670 Solution: Identify all fixed costs: venue fee $450, and sound equipment flat rate $220. Total fixed cost = 450 + 220 = 670. Variable cost for security guards: each guard costs $85, so for g guards, cost = 85g.
   Full step-by-step solution

   Step 1: Identify all fixed costs: venue fee $450, and sound equipment flat rate $220. Total fixed cost = 450 + 220 = 670. Step 2: Variable cost for security guards: each guard costs $85, so for g guards, cost = 85g. Step 3: Variable cost for sound equipment: $15 per hour, so for h hours, cost = 15h. Step 4: Total cost C is the sum of all fixed and variable costs: C = 85g + 15h + 670. Final answer: C = 85g + 15h + 670.

7. Sophia is designing a rectangular garden with a perimeter of 36 meters. The length is 6 meters more than the width. Write an equation in two variables for the perimeter. Answer: 2(l + w) = 36 Solution: When creating equations from geometric contexts, identify the key measurements and their relationships.
   Full step-by-step solution

   When creating equations from geometric contexts, identify the key measurements and their relationships. The perimeter formula for rectangles uses both length and width, and additional information about how these dimensions relate to each other helps create a complete mathematical model.

8. 2x² - 5x + 3 = 0 Answer: x = 1, x = 1.5 Solution: 2x² - 5x + 3 = 0 a x² + b x + c = 0 Here: a = 2, b = -5, c = 3 We look for two numbers that multiply to a * c = 2 * 3 = 6 and add to b = -5. The numbers -2 and -3 multiply to 6 but add to -5? No: (-2) + (-3) = -5, yes.
   Full step-by-step solution

   We are solving the quadratic equation: 2x² - 5x + 3 = 0 --- **Step 1: Identify coefficients** The equation is in the form: a x² + b x + c = 0 Here: a = 2, b = -5, c = 3 --- **Step 2: Check if factoring is possible** We look for two numbers that multiply to a * c = 2 * 3 = 6 and add to b = -5. The numbers -2 and -3 multiply to 6 but add to -5? No: (-2) + (-3) = -5, yes. So we split the middle term -5x into -2x - 3x. --- **Step 3: Factor by grouping** 2x² - 5x + 3 = 2x² - 2x - 3x + 3 Group terms: (2x² - 2x) + (-3x + 3) Factor each group: 2x(x - 1) - 3(x - 1) Now factor out (x - 1): (x - 1)(2x - 3) = 0 --- **Step 4: Solve for x** Set each factor equal to 0: x - 1 = 0 → x = 1 2x - 3 = 0 → 2x = 3 → x = 3/2 = 1.5 --- **Final Answer:** x = 1, x = 1.5